Derivative markets continue to evolve. Initial derivatives trading was for hedging purposes and essential for more efficient functioning of the financial system and the economy. Volumes for derivative markets grew with the need for financial capital enhancement as a substitute for reserves. While capital buffer regulatory requirements diminished the second function significantly, market-making regulations have put a spotlight on the former, particularly how to account for hedge book positions.

In spite of its importance, hedge book accounting is little understood. Shareholder concerns about the impact of a hedge book on equity.is intense. This post sheds some light on the subject from a balance sheet perspective as well an equity capital view. The essential point is that leveraged assets and liabilities are valued given the net present value of future cashflows. A hedge on assets is a contra-revenue event, designated as a contra-asset account. A hedge on funding costs or other liabilities is a contra-liability account. We integrate both of these derivative activities with in a balance sheet context, leaving aside the dynamic mark-to-market implementation.

First, some history. I suppose if we were to start at the source, it would be with de Finetti in the 1930s. He rewired probability as a degree of belief measure. As a thought-example, consider the statement “It is going to rain tomorrow.” Under the Aristotelian true/false dichotomy, it is a logically valid statement that has no truth content. Instead of a binary choice, de Finetti (1931, 1937) formulated an infinity of real-valued choices, normalized to the closed interval 0 and 1. Black and white was replaced by infinite shades of degree, based upon all available information, weighted by the past reliability of all known perspectives, and updated with prior and posterior knowledge. This is the conception of probability that gives contingent claims that link current states of the world to future stets of the world with rigor and consistency.

So far, so good in demonstrating the feasibility of derivative contracts. To make them functional, one must establish a fair price. To price a derivative, an axiomatic simplification is needed with respect to current and future states of the world: market completeness in Arrow and Debreu (1954). Market completeness, the assumption that all states of the world can be spanned by some combination of assets, enabled pricing for contingent claims in Black and Scholes (1973) and Merton (1973), indicating that derivative pricing can be based on replication arguments. No arbitrage and risk neutral valuation measures were outlined in Modigliani and Miller (1958) and introduced contingent claims into funding strategy and liability management. For those interested in a deeper dive, see Harrison and Pliska (1981), Delbaen and Schachermayer (2005) who advanced the theory, and the survey in Duffie (2010).

Modigliani and Miller (1958) and subsequent results depend on remarkable simplifications from reality. Derivative valuation adjustments such as cost of capital and cost of funding are zero. There is no justification for shareholders' capital at risk as a loss absorbing buffer. Completeness implies all trades are perfectly replicable, each and trade is valued at its price of replication, and independent of the endowment and of any other entity specific information.

The next step in the evolution is recognition of these deficiencies. The financial crisis of 2008 made clear that the market making function is at the core of the financial system. It forced an understanding that a hedge is not the same as capital buffers because of basis risk and counterparty risk. Counterparty and derivative valuation adjustments must be an explicit part of understanding bank balance sheets from a regulatory or investor perspective.

Recall two propositions:

*Proposition: The price of a “risk free” security = price of risky security + price of credit default swap + price of interest rate swap + price of FX swap.*

*The resulting quasi risk-free asset may well trade at par with treasury bills under ordinary market conditions, but they will gap wide during times of crisis.*

*Proposition: The process does not create a risk-free asset because each swap contract carries with it counterparty risk and variable valuation adjustments.*

*This gap, or basis risk, means that derivative contracts cannot shelter parties and counterparties from market realities even if the explicit terms seem to provide such a shield. *

Definition: Counterparty risk is the financial risk arising as a consequence of client or own default.

The risk of financial loss as a consequence of client default is hard to replicate since single name CDS instruments are illiquid and are typically written on bonds, not on swaps with rapidly varying value. Own default of the bank is even harder (not to say impossible) to hedge since, to hedge it, a bank would need to be able to freely trade its own debt. Counterparty credit risk is related to cash-flows or valuations linked to either counterparty default or the default of the bank itself.

Valuation adjustments are needed because there is a distinction between the fair valuation and entry price of a derivative contract. The discounted expectation of losses due to counterparty or own default are respectively known as CVA (credit valuation adjustment) and DVA (debt valuation adjustment). Regulators privilege collateralization as the form of counterparty risk mitigator.

The discounted expectations of the cost of funding cash collateral which can be rehypothecated is known as funding valuation adjustment (FVA), while the cost of funding segregated collateral to be posted as initial margin is the margin valuation adjustment (MVA).

The CVA, FVA and MVA flow into a reserve capital account (RC), which is held against expected counterparty default and funding losses. As losses are realized, reserve capital may deviate from its theoretical equilibrium level given by the expectation of future losses, a value we call target reserve capital (TRC). Banks are required to hold economic capital (EC) sufficient to absorb exceptional losses. Economic capital is typically computed to be the expected shortfall with confidence level 97.5% of the one year-ahead loss.

When actual costs differ radically form the prices determined by such formula in contracts, pressures arise to change the formula. Sellers cannot survive with prices below costs, and buyers are unwilling to pay prices grossly in excess of cost.

Proposition: Neither side can economically tolerate derivative prices radically different from prevailing market rates.

When market prices show a persistent tendency to differ radically from those determined by formulas in contracts, formula-based contracts are abandoned. Contracts are ended with a penalty fee, contracts are shortened, or the contract price is rebased to something better reflective of price dynamics.

These conditions make clear that the assumptions of the Modigliani-Miller theorem do not hold. The reulst is that shareholders' decisions in general depend on the funding strategy of the bank.

Let's begin. Recall the basic entity:

The assets of a bank consist of reserve capital (RC), gross shareholder capital (capital at risk SCR and uninvested equity), risk margins (KVA), derivative receivables and hedges of derivative payables. Its liabilities are debt, derivative payables and hedges of derivative receivables. Contra-assets and contra-liabilities are respective assets and liabilities deductions, which play a key role in bank valuation. While insurance portfolios have only a KVA-like metric called risk margin (RM), banks have several other metrics such as FVA and MVA that are related to funding the collateral involved in OTC derivative transactions.

Liquidation and funding costs are primarily driven by counterparty risk, and hard to hedge in practice.

Equity consists of reserve capital (RC) used for dealing with expected liquidation and funding costs as they occur. Exceptional losses are accounted for by economic capital (EC), which is a loss-absorbing resource devoted to cope with exceptional losses beyond reserve capital. EC consists of the sum between shareholders' capital at risk (SCR) and retained earnings (KVA).

Earnings received from clients to remunerate capital are accounted for as day-one profits, so they are immediately distributable, decided by the bank board of

directors.

Net Present Value

A defaultable entity is characterized by leverage, so that assets and liabilities are evaluated based on amortized net present value of future positive (on the asset side) and negative (on the liability side) cashflows.

Net present value (NPV) of any financial instrument can be established by dividing the expected net future receipts (that is, receipts less payments) associated with the instrument by a relevant discount factor.

PV = FA / (1 + I)n

in which:

PV = present value

FA = future amount receivable or payable

I = the appropriate interest rate

N= number of periods before amount becomes due.

NPV = SUM(PVR) – SUM(PVP)

in which:

PVR = present value of future amounts receivable

PVP = present value of future mounts payable.

Remark: The OIS rate is the best market proxy for a risk-free rate and the reference rate for the remuneration of cash collateral.

These equations require an adjustment to accommodate the hedge book and asset valuation adjustments.

Assets and contra-assets, liabilities and contra-liabilities, FVA (funding variation adjustment) and MVA (margin variation adjustment) further alter the equation:

Where

UCVA + FVA + MVA = Derivative receivables and hedges of derivative payables

FTDDVA + CVACL +FDA + MDA = Derivative payables and hedges of derivative receivables

UCVA is the unilateral CVA pricing the cash-flows valued by the CVA over an infinite time horizon

FTDDVA is the first-to-default DVA pricing the cash-flows valued by the DVA until the first default time of the bank and each considered counterparty

FTDCVA is the first-to-default CVA pricing the cash-flows valued by the CVA until the first default time of the bank and each considered counterparty

CVACL is the difference (UCVA - FTDCVA);

FDA (akin to the DVA2 in Hull and White (2012)) and MDA are further contra-liabilities respectively equal to the FVA and the MVA

AE is the accounting equity of the bank.

This can be simplified due to the definition of CVACL and if the funding terms cancel out:

Remark: Dealers are market makers and, as such, they are price makers. Dealer clients are price takers.

*Proposition: In an asymmetric setup with a price maker and a price taker, the price maker passes his costs to the price taker.*

Remark: In the case of bilateral trades between two dealers, each party will try to have the other pay its costs. This could result in either no deal or a shared loss somewhere in the middle of the range, depending on which party has the strongest contractual power.

**Shareholder Equity Perspective**

A defaultable entity has at least two different classes of stakeholders: shareholders and creditors. Shareholders have the control of the firm and are solely responsible for investment decisions up until the time of default. At default time, shareholders are wiped out. Creditors have no decision power until the time of default, but are protected by laws such as pari-passu forbidding certain trades that would trigger wealth transfers from them to shareholders.

Hence, there are two distinct but intertwined sources of market incompleteness:

Counterparty risk and market risk cannot be perfectly replicated,

Managers cannot offset wealth transfers from shareholders to bondholders and shareholders cannot realistically acquire all bank debt.

The corresponding contra-liabilities myopic, shareholders-centric balance sheet equation of a bank is:

Core Equity Tier I capital (CET1) is the metric meant to represent the fair valuation of shareholder capital.

Under Basel III, equity is targeted. Under the Total Loss-Absorbing Capacity (TLAC) standard, debt is converted into equity to replenish CET1. Changes in equity are measured by the formula:

**References**

Albanese, Claudio, Simone Caenazzo, and Stephane Crepey, “Capital Valuation Adjustment and Funding Valuation Adjustment”, https://arxiv.org/abs/1603.03012 accessed November 2018.

Arrow, K. J. and G. Debreu (1954). Existence of an equilibrium for a competitive economy. Econometrica 22 (3), 265{290.

Black, F. and M. Scholes (1973). The pricing of options and corporate liabilities. Journal of Political Economy 81, 637-59.

de Finetti, B. (1931). Sul significato soggettivo della probabilita. Fondamenta Mathematicae, 298-329.

de Finetti, B. (1937). La prevision: Ses lois logiques, ses sources subjectives. Annales de l'Institut Henri Poincare.

Delbaen, F. and W. Schachermayer (2005). The Mathematics of Arbitrage. Springer Finance.

Duffe, D. (2010). Dynamic Asset Pricing Theory. Princeton University Press.

Harrison, J. M. and S. R. Pliska (1981). Martingales and stochastic integrals in the theory of continuous trading. Stochastic Processes and their Applications, 215-260.

Merton, R. (1973). Theory of rational option pricing. Bell Journal of Economics and Management Science 4, 141-183.

Modigliani, F. and M. Miller (1958). The cost of capital, corporation finance and the theory of investment. Economic Review 48, 261-297.